Modular Form Data by Ken-ichi SHIOTA

Summary

I have calculated the follwing data on the spaces M- The (q,∞)-quaternion algebra D over
**Q** - A maximal order O in D
- The class number H and the type number T of D
- H is equal to the dimension of M
_{2}(Γ_{0}(q)). - T is equal to the dimension of the minus-eigenspace M
_{2}^{-}(Γ_{0}(q)) of the Atkin-Lehner involution.

- H is equal to the dimension of M
- The representatives { A
_{j}}_{ j = 1,...,H}of the left O-ideal classes - The maximal orders O
_{j}= (A_{j})^{-1}A_{j}in D - The representatives { A
_{ij}}_{ i = 1,...,H}of the left O_{j}-ideal classes- A
_{ij}is obtained by A_{ij}= ( a constant multiple of ) (A_{j})^{-1}A_{i}.

- A
- The group order e
_{j}of the unit group of O_{j} - The theta series θ
_{ij}associated to the norm form of A_{ij}- The definition of the theta series is

θ_{ij}(z) = (1/e_{j}) Σ_{x in Aij}exp(2πiz N(x)/N(A_{ij}))

where N denotes the reduced norm on D. - We have e
_{j}θ_{ij}= e_{i}θ_{ji}, hence we have to calculate θ_{ij}only for i >= j.

- The definition of the theta series is
- The Brandt matrices B(n) ( n = 0,1,2,... )
- The definition of the Brandt matrices is

Σ_{n = 0,1,2,...}B(n) exp(2πinz) = ( θ_{ij}(z) )_{ i,j = 1,...,H} - For n >= 1, B(n) is an integral representation matrix of the Hecke operator T(n) on M
_{2}(Γ_{0}(q)).

- The definition of the Brandt matrices is
- The characteristic polynomial of B(n) ( = that of T(n) ) and its factorization over
**Z**

- The common eigenvectors of B(n)'s, and the eigenvalues of B(n
_{0}) for a selected n_{0}

- Each eigenvector corresponds to the Eisenstein series E or a newform f in the space of the elliptic cusp forms S
_{2}(Γ_{0}(q)). - Each eigenvalue is equal to a Hecke eigenvalue, hence to a Fourier coefficient of E or f.

- Each eigenvector corresponds to the Eisenstein series E or a newform f in the space of the elliptic cusp forms S

Tables

Reference

- [HPS] Hiroaki Hijikata, Arnold K. Pizer and Thomas R. Shemanske, The basis problem for modular forms on Γ
_{0}(N), Mem. Amer. Math. Soc. 82 (1989), no. 418. - [Pi80]
A. Pizer, An algorithm for computing modular forms on Γ
_{0}(N), Journal of Algebra 64,(1980), 340-390. - [Shio91]
Ken-ichi Shiota, On theta series and the splitting of S
_{2}(Γ_{0}(q)), J. Math. Kyoto Univ., 31 (1991.12), 909-930.

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